{"id":"n-8217f7f3de935aeb","revision":1,"first_index":0,"fingerprint":"f6-9a4b425d2a954ae8","name":"basis_map_A013609 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","4"],["11","30","20"],["45","186","252","112"],["197","1092","2240","2016","672"],["903","6280","17280","23520","15840","4224"],["4279","35802","123660","225720","229680","123552","27456"],["20793","203350","845460","1937320","2642640","2146144","960960","183040"],["103049","1153416","5608064","15471456","26490464","28828800","19475456","7468032","1244672"],["518859","6540876","36414144","117507936","242240544","330849792","299400192","173122560","58046976","8599552"],["2646723","37106550","232754340","860212080","2074450560","3411023616","3873192960","2999093760","1515671040","451476480","60196864"],["13648869","210649362","1469977740","6122534880","16911861120","32530891008","44466960384","43195384320","29224419840","13115970560","3514570752","426008576"],["71039373","1196815500","9196640640","42623254080","132701722560","292391548416","467536143360","546668559360","463906967040","278655252480","112466264064","27386265600","3042918400"],["372693519","6805767696","57102948864","291483356736","1009947242304","2508342051072","4594340754432","6283714145280","6417731713536","4834375749632","2610823987200","957302128640","213612871680","21909012480"],["1968801519","38735977266","352366829724","1964305737576","7497022316784","20723659276320","42787334606016","67032497930112","80076419334912","72589065669120","49154200775680","24111552890880","8099488051200","1667823575040","158840340480"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","61","595","6217","68047","770149","8939707","105843409","1273241431","15517824973","191202877411","2377843390873","29807864423071","376255282112629"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"],["6","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166568,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f033b6e7de8ef377"]},{"id":166569,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-f033b6e7de8ef377"]},{"id":166570,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-f033b6e7de8ef377"]},{"id":166571,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-f033b6e7de8ef377"]}],"region_exact":{},"incoming_edges":[{"id":"e-f033b6e7de8ef377","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-8217f7f3de935aeb","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"0ab91a86f8e111947c7219426bfabe6ac2758a507280b5e1d542eb58db128a7a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-b1b90b75d99861dc","kind":"oeis_match","node_a_id":"n-8217f7f3de935aeb","node_b_id":null,"fingerprint":null,"oeis_id":"A269731","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A269731","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A269731","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-8217f7f3de935aeb","oeis_id":"A269731","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":23261}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      3,       4;\n  2:     11,      30,       20;\n  3:     45,     186,      252,       112;\n  4:    197,    1092,     2240,      2016,       672;\n  5:    903,    6280,    17280,     23520,     15840,      4224;\n  6:   4279,   35802,   123660,    225720,    229680,    123552,     27456;\n  7:  20793,  203350,   845460,   1937320,   2642640,   2146144,    960960,    183040;\n  8: 103049, 1153416,  5608064,  15471456,  26490464,  28828800,  19475456,   7468032,  1244672;\n  9: 518859, 6540876, 36414144, 117507936, 242240544, 330849792, 299400192, 173122560, 58046976, 8599552;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(n-1)*P(n-2,x) + (6*n+3+(8*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 1)*P[n - 2, x] + (6*n + 3 + (8*n + 4)*x)*P[n - 1, x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1)*v[i - 2] + (6*n + 3 + (8*n + 4)*x)*v[i - 1])/(n + 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-66e2c3448245b0b7","steps":["basis A013609"],"edges":["e-f033b6e7de8ef377"],"text":"A033282 \u2192 basis A013609"},"headline":"A033282 \u2192 basis A013609","chain":null}